INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 12, DECEMBER 2013

ISSN 2277-8616

Some Applicable Methods Of Approximating Basic
Trigonometric Functions And Their Inverse Value
Manaye Getu Tsige
Abstract: This paper presents some applicable methods of approximating basic trigonometric functions and their inverse value. Methods are the best
choice when a need arise to know first few digits after a decimal point and corresponding angle without spending time for immediate purpose. The ways
of approximation are helpful for science and engineering field of study; they can be applied to get immediate solutions for practical problems which might
be estimating, comparing and judging while operations of numbers. The assumption stated to carry out this work is; There exists certain function which
can satisfies the condition defined as; if the sequence of some domain values within a domain of function forms an arithmetic progression, then the
sequence of corresponding range values within a range of function will also forms an arithmetic progression. This assumption leads to the assumed
generalized approximate equation and finally to the major findings. The major areas of study to carry out this particular work are arithmetic progression,
sine function, cosine function and idea related to trigonometric functions such as trigonometric identities, co terminal angles, reference angle and co
function definition. The objective is to contribute additional alternative knowledge to the Mathematical science. The findings of this paper are useful to
derive general approximation formulae and other related findings that will be presented in the future.
Index Term: sine function, cosine function, co function, trigonometric identities, arithmetic sequence.
————————————————————

1

INTRODUCTION

Trigonometric functions arise in geometry but they are also
applicable to study sound, the motion of pendulum and
many other phenomena involving rotation and oscillation
[4]. Different approximation techniques have been used to
approximate trigonometric functions since along ago. For
instance, evaluating trigonometric functions of common
angles and quadrant angles by using the properties of
triangle and unit circle respectively [10],[7]. Trigonometric
table is often used to approximate value and inverse value
of trigonometric functions. Sometimes it is possible to use
trigonometric identities, sum of two angle formulae, double
angle formulae, half angle formulae, product to sum
formulae and sum to product formulae provided that angle
is reduced to common angles[10] unless otherwise they
have no advantage upon evaluating trigonometric functions.
There are equations known as Maclaurin series They are
given by [9],[4]

Above equations are also listed in different books
[3],[11],[1],[13],[5]. Small angle approximation is given by;
for very small x,
[4]. This paper provides the best
alternative methods that should be chose when a need
arise to know first few digits after a decimal point and
corresponding angle without spending time for immediate
purpose.

Methods are helpful for science and engineering field of
study; they can be applied to get immediate solutions for
practical problems which might be estimating, comparing
and judging while operations of numbers. The objective of
this paper is to contribute additional alternative knowledge
to the Mathematical science. The major areas of study to
carry out this particular work are arithmetic progression,
sine function, cosine function and idea related to
trigonometric functions such as co terminal angles,
reference angle and co function definition. Author Hornsby
et al., [6] discusses ideas related to trigonometric functions.
Co function definition for sine and cosine function is given
by

In other word, if

then

Authors (Larson and Hosteller [10], Bronshtein, et al., [2],
Solomon [12] discuss about arithmetic sequence. If the
difference between successive terms of a sequence is
constant, then the sequence is an arithmetic progression.
Let
is an arithmetic progression, then

If the arithmetic progression has first term
and common
difference d, then the nth term of the sequence given by

2

MATERIALS AND METHODS

Use the following assumption to arrive at assumed
generalized approximate equation and use this equation as
starting idea to derive provable methods and finally to other
related findings.

Assume sine function approximately
condition within the given interval

Use this equation as starting idea to derive provable
methods and those methods collectively named as some
applicable methods.

satisfies

above
Let

Substitute Eq (2) into (3)

For each value of within the interval
following inequality holds true.

Substitute Eq (1) into (3)

Substitute Eq (4) into (5)

For non-negative increasing terms of an arithmetic
progression, the following inequality holds true.

Having this interval in mind, Let
it into Eq (6)

and substitute

, the

Decimal representation of real numbers: Authors
Wrede and Spiegel [14] discuss about decimal
representation of real numbers. In order to equally express
real number in terms of decimal number, use a few digits
and dotted line after a decimal point;
Consider decimal representation of real numbers
and introduce variables to represent digits after a decimal
point.
Definition: To equally express
in terms of decimal
number, known digits after a decimal point are denoted by
variables
while other unknown digits are
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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 12, DECEMBER 2013

Table 1. Digits versus corresponding sine value

8

9
0.09

7

0.08

6

0.07

5

0.06

4

0.05

3

0.04

2

0.03

1

0.02

0

0.01

A
B x
0

0.00

denoted by using dotted line. Having this definition in mind,
consider two digits after a decimal point and let digits and
are known digits after a decimal point, then
approximated by

ISSN 2277-8616

5

0.19
0.29
0.39
0.49

0.18
0.38
0.48

0.28

0.17
0.37
0.47

0.27

0.16
0.36
0.46

0.26

0.15
0.35
0.45

0.25

0.14
0.34
0.44

0.24

0.13
0.33
0.43

0.23

0.12
0.32
0.42

0.22

0.11
0.21

0.20

0.31

4

0.41

Substitute Eq (10) into (8)

0.30

3

0.40

2

Correlate Eqs (9), (10) and (11)

0.50

1

0.10

Substitute Eq (10) into (9)

Substitute Eq (13) into (12)
Table 1 shows Eq (15) holds true for all two digits decimal
numbers. Equations (8) and (16) satisfy co function
definition.
If the other unknown digits are taken into account and
denoted by dotted line, then the approximate Eq (14) can
be written as the Eq (15)

Mind above definition; variables
and
stand for digits
after a decimal point where as dotted line stands for other
unknown digits. There are 51- two digits decimal numbers
within the interval
and hence it is now very
easy to verify equation (15) whether it holds true or not for
all possible two digits decimal numbers. Use trigonometric
table to crosscheck equation (15) for each two digits
decimal number.

If the negative and other co terminal angles are taken into
account, then the Eq (15) can be modified as the method
one and state the equations (8) and (16) as the first
approximation
method.
The
maximum
error
of
approximation can be calculated by using known digits after
decimal point.
2.1.1 Method one
If the
stand for digits after a decimal point and the
angle is expressed in degree unit, then the following
equations expressed in decimal number hold true.

Let
Where n is integer
The maximum error of approximation is less than 0.009
2.1.2 First approximation method
Consider co function definition [7]

Summarize it by constructing table for all two digits decimal
numbers (0. AB)

The maximum error of approximation is less than 0.009 All
other methods can be derived in the same way as above
methods derived and each method can be verified in the
same manner. And therefore, it is unnecessary to repeat
steps hereafter. Take
to derive method two and
second approximation method.
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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 12, DECEMBER 2013

2.1.3 Method two
If the
stand for digits after a decimal point and the
angle is expressed in degree unit, then the following
equations expressed in decimal number hold true.

ISSN 2277-8616

2.1.7 Method four
If the
stand for digits after a decimal point
and the angle is expressed in degree unit, then the
following equations expressed in decimal number hold true.

Equations hold true for all 17453 six digits decimal numbers
within the interval;
The
maximum error of approximation is less than 0.0000009
2.1.8 Fourth approximation method
Equations hold true for all 259 three digits decimal numbers
(
within the interval;
The
maximum error of approximation is less than 0.0009
2.1.4 Second approximation method
The maximum error of approximation is less than
0.0000009

3

The maximum error of approximation is less than 0.0009
Next, take
to derive method three and third
approximation method

RESULTS AND DISCUSSION

The main results of this paper are method one, two, three,
four and corresponding approximation methods. They are
collectively named as some applicable methods and they
are used to compute value and inverse of basic
trigonometric functions. For instance, in the case of inverse
value computation, method one, two and three can also be
written by

2.1.5 Method three
If the
stand for digits after a decimal point and
the angle is expressed in degree unit, then the following
equations expressed in decimal number hold true.

Equations hold true for all 872 four digits decimal numbers
within the interval;
The maximum
error of approximation is less than 0.00009
2.1.6 Third approximation method

Within common interval, the error of approximation gets
smaller and smaller from the first up to the fourth
approximation method. This implies that the assumed
generalized approximate equation is very interesting and
supposed to be holds true. And therefore, Sine function
approximately satisfies above condition within the interval;
provided that the negative angles are taken into
consideration.

3.1 APPLICATIONS
The results of this research work have applications in
Engineering, Military institution, Education.
The maximum error of approximation is less than 0.00009
Next, take
to derive method four and fourth
approximation method

3.1.1 Military institution/Engineering areas
To set angle based on target of interest. Compares to all
other trigonometric functions evaluation techniques found in
literature, this paper contains expression in the form of
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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 12, DECEMBER 2013

ISSN 2277-8616

simple fraction that it enables to set angle by using a value
of numerator and denominator.
Example 1: Target direction of fire at
but not exactly
)

(near to

4

CONCLUSIONS

In general, the limitation of this paper is maybe
approximation becomes possible only within the given
interval. It provides foot step to derive general
approximation formulae and other related findings that will
be presented in the future.